Theorem gennfrw-P6 685

Description: General for ⇒ ENF in (weakened form).

Requires the existence of '𝜑₁(𝑥₁)' as a replacement for '𝜑(𝑥)'.

If '𝜑' is general for '𝑥', then '𝑥' is effectively not free in '𝜑'.

Hypotheses

Ref	Expression
gennfrw-P6.1	⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
gennfrw-P6.2	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
gennfrw-P6	⊢ Ⅎ𝑥𝜑

Distinct variable groups:   𝜑,𝑥₁   𝜑₁,𝑥   𝑥,𝑥₁

Proof of Theorem gennfrw-P6

Step	Hyp	Ref	Expression
1		gennfrw-P6.2	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	alloverimex-P5.RC.GEN 603	. . 3 ⊢ (∃𝑥𝜑 → ∃𝑥∀𝑥𝜑)
3		gennfrw-P6.1	. . . 4 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
4	3	exgenallw-P6 680	. . 3 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)
5	2, 4	syl-P3.24.RC 260	. 2 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
6		dfnfreealt-P6 683	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
7	5, 6	bimpr-P4.RC 534	1 ⊢ Ⅎ𝑥𝜑

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by:  nfrv-P6  686  nfrall1w-P6  692  nfrex1w-P6  693  nfrall2w-P6  694  nfrex2w-P6  695  exgennfrw-P6  697